Question

Find the value of a if the three equations are consistent
$(a+1{)}^{3}x+(a+2{)}^{3}y=(a+3{)}^3$
$(a+1)x+(a+2)y=a+3$
$x+y=1$.

Answer 1

Since the given equations are consistent,

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$......(a)

Considering the equations,

$(a+1{)}^{3}x+(a+2{)}^{3}y=(a+3{)}^3$......(1)

$(a+1)x+(a+2)y=(a+3)$.....(2)

From (a)

$\frac{(a+1{)}^3}{(a+1)}=\frac{(a+2{)}^3}{(a+2)}=\frac{(a+3{)}^3}{(a+3)}$

Considering 1st and 2nd terms, we have,

$\frac{(a+1{)}^3}{(a+1)}=\frac{(a+2{)}^3}{(a+2)}$

$(a+1{)}^2=(a+2{)}^2$

$a^2+2a+1=a^2+4a+4$

$-3=2a$

$\Rightarrow a=\frac{-3}{2}$